Every finite set of natural numbers is realizable as algebraic periods of a Morse$\unicode{x2013}$Smale diffeomorphism

TL;DR

The paper solves the realizability problem for algebraic periods by proving that any finite set ${ m A}$ can be realized as the algebraic period set ${ m AP}(f)$ of a Morse–Smale diffeomorphism on some closed surface, with explicit genus formulas for orientable, non-orientable, and orientation-reversing cases. The construction is geometric: it builds surface pieces carrying periodic maps, glues them to control Lefschetz data via Möbius inversion, and ensures Morse–Smale realizability through da Rocha’s theorem; a no-odd-period restriction appears in the orientation-reversing case. The work also links ${ m AP}_{ m odd}$ with the minimal Lefschetz period set ${ m MPer}_L(f)$, analyzes the Nielsen–Thurston classifications, and provides a lower bound on the number of conjugacy classes of algebraically finite type mapping classes that grows exponentially with genus. Together, these results advance understanding of how algebraic invariants constrain, and are constrained by, topological dynamics on surfaces, with implications for periodic-point theory and mapping-class enumeration.

Abstract

A given self-map $f\colon M\to M$ of a compact manifold determines the sequence $(L(f^n))$ of the Lefschetz numbers of its iterations. We consider its dual sequence $(a_n(f))$ given by the Möbius inversion formula. The set ${\mathcal AP}(f)=\{ n\in \mathbb N\ \colon\ a_n(f)\neq 0\}$ is called the set of algebraic periods. We solve an open problem existing in literature by showing that for every finite subset ${\mathcal A}$ of natural numbers there exist an orientable surface $S_{\rm g}$, as well as a non-orientable surface $N_{\rm g}$, of genus ${\rm g}$, and a Morse$\unicode{x2013}$Smale diffeomorphism $f$ of this surface such that $\mathcal{AP}(f)=\mathcal{A}$. For such a map it implies the existence of points of a minimal period $n$ for each odd $n \in \mathcal{A}$. For the orientation-reversing Morse$\unicode{x2013}$Smale diffeomorphisms of $S_{\rm g}$, we identify strong restrictions on ${\mathcal AP}(f)$. Our method also provides an estimate of the number of conjugacy classes of mapping classes containing Morse$\unicode{x2013}$Smale diffeomorphisms, which is exponential in ${\rm g}$.

Figures (2)

• Figure 1: Orientation-preserving case.
• Figure 2: Orientation-reversing case.

Theorems & Definitions (34)

• Definition 1.1
• Example 1.2
• Theorem 1.3
• Theorem 2.1: Nielsen--Thurston
• Proposition 2.2
• Theorem 2.3: L. F. da Rocha, 1985
• Remark 2.4
• Proposition 2.5
• proof
• Theorem 3.1